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Minimal competencies for CSC 133 – Approved 4/14/00
Discrete Structures
Logic
 Compound Statements: Demonstrate understanding of the logical connectives
conjunction, disjunction and negation by applying logical equivalence laws to reduce
complex expressions.
 Quantified Statements: Recognize valid argument forms including the rule of
universal instantiation, universal modus ponens and universal modus tollens and be
able to correctly identify converse and inverse errors.
 Compound Quantified Statements: Write the negation of compound quantified
statements.
Elementary Number Theory
 Use the division algorithm to prove parity of the integers.
 Solve problems using integer division and modulo arithmetic.
 Recognize instances of and solve problems using the sum of the first n integers and
the sum of a geometric sequence.
 Use logarithms, floor, and ceiling notation in calculations.
 Demonstrate proficiency in matrix arithmetic.
Sequences
 Demonstrate proficiency with summation and product notation including
transformation by change of variable.
Methods of Proof
 Write simple proofs of theorems relating to elementary number theory utilizing
several of the following methods: Direct Proof, Disproof by Counter Example,
Indirect Arguments using either Contraposition or Contradiction, and Mathematical
Induction.
 Use Mathematical Induction to prove the validity of an explicit formula for a
recurrence relation.
Set Theory
 Know the definitions of equality, union, intersection, set difference, and complement
of sets, empty set, power set, set partition, and Cartesian products.
 Distinguish between "element of" and "subset of".
 Use logical equivalencies to simplify complex set statements.
Boolean Algebra
 Incorporate an axiomatic basis of Boolean Algebra, including DeMorgan's Theorem
in problem solving.
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Utilize Canonical Forms: Product Form, Sum Form, Minterms, and Maxterms.
Demonstrate familiarity with design aids: Karnaugh maps and the Quine-McClusky
method.
Implement logic circuits using AND/OR and NAND/NOR circuits..
Recursion
 Demonstrate an understanding of recursion using prototypical examples: Towers of
Hanoi, Fibonacci numbers, and factorials.
 Solve recurrence relations iteratively.
 Find an explicit formula for second order linear homogeneous recurrence relations
given the characteristic equation, single and double root theorems.
Graphs and Trees
 Know basic definitions of multigraph, simple graph, digraph, bipartite graphs, walks,
paths, circuits, subgraph, connected graph, and graph components.
 Represent graphs using matrices. Identify basic graph properties in matrices: n-walks,
degrees of vertices, in-degree, out-degree, and connectedness.
 Determine isomorphism for small graphs or use isomorphic invariants to show two
graphs are not isomorphic.
 Identify Eulerian graphs.
 Be familiar with basic properties of trees, rooted trees, binary trees, full binary trees
and m-ary trees.
 Determine the feasibility of basic graphs given properties such as order, size,
cardinality of internal or terminal vertices, and score sequence.
 Use either Kruskal's or Prim's algorithm to find a minimum spanning tree.
Relations
 Define relations on sets, binary relations, and m-ary relations.
 Identify Reflexive, irreflexive, symmetric, asymmetric, and transitive relations using
digraphs, adjacency matrices, and or set elements as applicable.
 Determine whether relations are equivalence or partial order relations.
 Identify equivalence classes of a relation defined on a finite set.
Counting
 Demonstrate understanding of sample space.
 Count elements of lists, sublists, and one-dimensional arrays.
 Distinction between finite and infinite sets
 Distinction between countable and uncountable sets
 Use the multiplication and addition rules.
 Use the formulae for permutations and combinations.
 Demonstrate proficiency using the inclusion/exclusion rule.
 Use the binomial theorem and Pascal's triangle.